Reversible, irreversible, and mixed regimes for periodically driven disks in random obstacle arrays

We examine an assembly of repulsive disks interacting with a random obstacle array under a periodic drive and find a transition from reversible to irreversible dynamics as a function of drive amplitude or disk density. At low densities and drives, the system rapidly forms a reversible state where the disks return to their exact positions at the end of each cycle. In contrast, at high amplitudes or high densities, the system enters an irreversible state where the disks exhibit normal diffusion. Between these two regimes, there can be an intermediate irreversible state where most of the system is reversible, but localized irreversible regions are present that are prevented from spreading through the system due to a screening effect from the obstacles. We also find states that we term “combinatorial reversible states” in which the disks return to their original positions after multiple driving cycles. In these states, individual disks exchange positions but form the same configurations during the subcycles of the larger reversible cycle.

Figure 1

Obstacle locations (red circles) and mobile disk locations (blue) along with trajectory lines indicating the net translation of the mobile disks from one cycle to the next over a time of $N_c=100$ driving cycles. The disk positions are shown at the end of the drive cycle when the drive is about to switch from the $-x$ direction back to the $+x$ direction. (a) A completely reversible state at $A=0.02$ and $\rho =0.368$, where all of the disks return to the same position at the end of each driving cycle. The trajectory lines have zero length and thus do not appear in the panel. (b) An irreversible state at $A=0.015$ and $\rho =0.579$. The trajectory lines are finite and disordered.

Figure 2

A sample with total disk density $\rho =0.398$ at varied drive amplitudes $A=0.01$ to $A=0.04$. (a) The distance $R(n=1)$ traveled by the disks in a single driving cycle vs the total number of elapsed driving cycles $N_c$, where we have set $t_0=N_c-1$ in the calculation of $R$. (b) The total distance traveled $d$ vs $N_c$. For $A=0.01$, 0.015, and 0.02, the system reaches a reversible state. For $A=0.025$, the system is irreversible but shows subdiffusive behavior, and for $A=0.03$, 0.035, and 0.04, the motion is irreversible with regular diffusion.

Figure 3

(a) Heat map of the fraction $f$ of the five different disorder realizations that reached reversible states plotted as a function of $\rho$ vs $A$. (b) Heat map of $ln\left[d(N_c=3000)\right]$, the total displacement measured during $N_c=3000$ driving cycles with $t_0=1000$ driving cycles, as a function of $\rho$ vs $A$.

Figure 4

Heat map of $n^{*}$, the smallest value of $n$ for which $R\left(n\right)$ reaches zero in the reversible regime, as a function of $\rho$ vs $A$. When $n^{*}>1$, the state is multicycle reversible. In the irreversible regime, we mark $n^{*}=0$. In the reversible regime, the value of $n^{*}$ is averaged only over disorder realizations that were reversible.

Figure 5

Heat map as a function of $\rho$ vs $A$ of the exponent $\alpha$ obtained from a fit to $d\propto N_c^{\alpha }$. $\alpha =0$ indicates reversible behavior, and $\alpha =1.0$ indicates diffusive behavior.

Figure 6

Obstacle locations (red circles) and mobile disk locations (blue) along with trajectory lines indicating the net translation of the mobile disks from one cycle to the next over a time of $N_c=100$ driving cycles in a state with two local regions of irreversible behavior but no long time diffusion at $\rho =0.519$ and $A=0.01$.

Figure 7

Illustration of a combinatorial reversible state at $\rho =0.337$ and $A=0.025$ where the obstacle locations (large red circles), nonexchanging mobile disks (white circles), and exchanging mobile disks (small colored circles) are shown in only a small portion of the sample. The small trajectory lines indicate the net distance moved by each disk compared with its position at the end of the previous driving cycle. The disks return to their original positions every twelve cycles. Each panel shows the disk configuration at the end of a drive cycle. Time increases from left to right and top to bottom so that the cycle sequence is A1-A2-A3-A4-B1-B2-B3-B4-C1-C2-C3-C4. The macroscopic disk configurations repeat every four drive cycles, so that A1, B1, and C1 have the same macroscopic disk configuration, but the individual disks are permuted within this configuration.

Figure 8

Illustration of a different small region of the combinatorial reversible state from Fig. 7 at $\rho =0.337$ and $A=0.025$, showing the obstacle locations (large red circles) and exchanging mobile disks (small colored circles). The small trajectory lines indicate the net distance moved by each disk compared with its position at the end of the previous driving cycle. The disks return to their original positions every 18 cycles. Each panel shows the disk configuration at the end of a drive cycle. Time increases from left to right and top to bottom. The macroscopic disk configurations repeat every six drive cycles, so that A1, B1, and C1 have the same macroscopic disk configuration, but the individual disks are permuted within this configuration.